We now describe the signaling scheme in mathematical terms. Let \(\mathcal {S}\)  be a finite set of signals, from which the leader draws a signal \(s\in \mathcal {S}\)  according to a conditional distribution specified by the scheme. Mathematically, a signaling scheme is a mapping \(\phi :[M]\times \Theta \rightarrow \Delta (\mathcal {S})\) , which determines the probability of sending signal \(s\)  conditional on the realized action \(i\in [M]\)  and the reported type \(\hat{\theta }\in \Theta \) , that is \(\mathbb {P}(s|i,\hat{\theta })=\phi (s|i,\hat{\theta })\) . To simplify our exposition, by revelation principle in Bayesian persuasion [1]}, we consider the direct signaling device, for which the signal space coincides with the follower's action space, that is \(\mathcal {S}=[N]\) . Under this signaling, each signal can be interpreted as an action recommendation. In the example in sec:example, for the type \(\theta _1\)  follower, the signal \(s_0\)  can be replaced by \(j_0\) , suggesting that the follower would better leave the market since the leader is devoting resources to the corresponding product. Similarly, \(s_1\)  is equivalent to \(j_1\) , which means the leader encourages the follower to enter the market. However, we note that this is merely an interpretation, and each signal, except inducing a posterior belief, does not carry a meaning beyond itself.
